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Solving combinatorial optimisation problems using oscillator based Ising machines

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Abstract

We present OIM (Oscillator Ising Machines), a new way to make Ising machines using networks of coupled self-sustaining nonlinear oscillators. OIM is theoretically rooted in a novel result that establishes that the phase dynamics of coupled oscillator systems, under the influence of subharmonic injection locking, are governed by a Lyapunov function that is closely related to the Ising Hamiltonian of the coupling graph. As a result, the dynamics of such oscillator networks evolve naturally to local minima of the Lyapunov function. Two simple additional steps (i.e., turning subharmonic locking on and off smoothly, and adding noise) enable the network to find excellent solutions of Ising problems. We demonstrate our method on Ising versions of the MAX-CUT and graph colouring problems, showing that it improves on previously published results on several problems in the G benchmark set. Using synthetic problems with known global minima, we also present initial scaling results. Our scheme, which is amenable to realisation using many kinds of oscillators from different physical domains, is particularly well suited for CMOS IC implementation, offering significant practical advantages over previous techniques for making Ising machines. We report working hardware prototypes using CMOS electronic oscillators.

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Notes

  1. Indeed, since the Ising Hamiltonian ground state problem is NP-complete (Barahona 1982), every NP-complete problem can in principle be mapped to Ising.

  2. This is somewhat analogous to annealing schedules that are used in simulated annealing (SA, Myklebust 2015), though we stress that the underlying minimization mechanism of OIM is completely different from that of SA.

  3. Moreover, as we show in Sect. 3.4, our scheme is inherently resistant to variability even without such calibration.

  4. More generally, \(c_{ij}\)s can be any \(2\pi \)-periodic odd functions, which are better suited to practical oscillators.

  5. In the Ising Hamiltonian (1), \(J_{ij}\) is only defined when \(i<j\); here we assume that \(J_{ij} = J_{ji}\) for all i,  j.

  6. More generally, we can use \(\{2k\pi\ |\ k\in {\mathbf {Z}}\}\) and \(\{2k\pi +\pi\ |\ k\in {\mathbf {Z}}\}\) to represent the two states for each oscillator’s phase.

  7. We ran simulated annealing for a long time (1m) multiple times, then treated the best results as global optima.

  8. Benchmarks G1\(\sim \)21 are of size 800; G22\(\sim \)42 are of size 2000; G43\(\sim \)47, G51\(\sim \)54 are of size 1000; G48\(\sim \)50 are of size 3000.

  9. Results and runtimes for SS, CirCut and VNSPR are available in the “Computational Experiences” section of Wang (2017).

  10. Note that the 200 simulations can be run in parallel to greatly reduce this runtime. However, we stress that software simulation time is not of immediate relevance for OIM; instead, it is the time the OIM hardware will take to solve the problem, as discussed below.

  11. The results here were re-generated using updated code, hence do not exactly match the results we reported previously (Wang and Roychowdhury 2019a, b).

  12. Each single loop has one frustrated edge by design. When they overlap, the weights of the frustrated edges can change, and they may not remain individually frustrated. This does not change the result regarding the global minimum energy level.

  13. Different criteria are used in different versions of frustrated loop benchmarks. In Sheldon et al. (2019), a loop is too short when its length is smaller than 6; in Hen et al. (2015), when its length is smaller than 8; in King et al. (2015), when the loop remains confined in a 8-vertex cube. We choose to be consistent with Sheldon et al. (2019) for this preliminary exploration.

  14. In Sheldon et al. (2019) and Hen et al. (2015), the frustrated edge is chosen randomly; in King et al. (2015), the frustrated edge is chosen randomly among edges with a total weight below a certain threshold, such that ideally most edge weights reach the threshold eventually.

  15. e.g., we have not yet explored OIM parameter tuning to achieve lower TTS.

  16. Ising machines can be used on general graph colouring problems, and this four-colouring problem is chosen here for illustrative purposes. Four-colouring a planar graph is actually not NP-hard and there exist polynomial-time algorithms for it Robertson et al. (1996).

  17. Hawaii and Alaska are considered adjacent, hence their colours will be different in the map.

References

  • Acebrón JA, Bonilla LL, Vicente CJP, Ritort F, Spigler R (2005) The Kuramoto model: a simple paradigm for synchronization phenomena. Rev Mod Phys 77(1):137

    Article  Google Scholar 

  • Aramon M et al (2019) Physics-inspired optimization for quadratic unconstrained problems using a digital annealer. Front Phys 7:48

    Article  Google Scholar 

  • Barahona Francisco (1982) On the computational complexity of Ising spin glass models. J Phys A Math Gen 15(10):3241

    Article  MathSciNet  Google Scholar 

  • Bhansali P, Roychowdhury J (2009) Gen-Adler: the generalized Adler’s equation for injection locking analysis in oscillators. In: Proceedings of IEEE ASP-DAC, pp 227–522

  • Bian Z, Chudak F, Israel R, Lackey B, Macready WG, Roy A (2014) Discrete optimization using quantum annealing on sparse Ising models. Front Phys 2:56

    Article  Google Scholar 

  • Bian Z, Chudak F, Macready WG, Rose G (2010) The Ising model: teaching an old problem new tricks. D-Wave Syst, 2

  • Brush Stephen G (1967) History of the Lenz-Ising model. Rev Mod Phys 39(4):883–893

    Article  Google Scholar 

  • Burer S, Monteiro R, Zhang Y (2002) Rank-two relaxation heuristics for max-cut and other binary quadratic programs. SIAM J Optim 12(2):503–521

    Article  MathSciNet  Google Scholar 

  • Camsari KY, Faria R, Sutton BM, Datta S (2017) Stochastic p-bits for invertible logic. Phys Rev X 7(3):031014

    Google Scholar 

  • Demir A, Mehrotra A, Roychowdhury J (2000) Phase noise in oscillators: a unifying theory and numerical methods for characterization. IEEE Trans Circ Syst I Fund Theory Appl 47(5):655–674

    Article  Google Scholar 

  • Denchev VS, Boixo S, Isakov SV, Ding N, Babbush R, Smelyanskiy V, Martinis J, Neven H (2016) What is the computational value of finite-range tunneling? Phys Rev X 6(3):031015

    Google Scholar 

  • Ercsey-Ravasz M, Toroczkai Z (2011) Optimization hardness as transient chaos in an analog approach to constraint satisfaction. Nat Phys 7(12):966

    Article  Google Scholar 

  • Festa P, Pardalos PM, Resende MGC, Ribeiro CC (2002) Randomized heuristics for the MAX-CUT problem. Optim Methods softw 17(6):1033–1058

    Article  MathSciNet  Google Scholar 

  • Gyoten H, Hiromoto M, Sato T (2018) Area efficient annealing processor for Ising model without random number generator. IEICE Trans Inform Syst E101.D(2):314–323

    Article  Google Scholar 

  • Gyoten H, Hiromoto M, Sato T (2018) Enhancing the solution quality of hardware Ising-model solver via parallel tempering. In: Proceedings of ICCAD’18, pp 70:1–70:8, New York, NY, USA, ACM

  • Hamerly R et al (2019) Experimental investigation of performance differences between coherent Ising machines and a quantum annealer. Sci Adv 5(5):eaau0823

    Article  Google Scholar 

  • Harris R, Johansson J, Berkley AJ, Johnson MW, Lanting T, Han S, Bunyk P, Ladizinsky E, Oh T, Perminov I et al (2010) Experimental demonstration of a robust and scalable flux qubit. Phys Rev B 81(13):134510

    Article  Google Scholar 

  • Helmberg C, Rendl F (2000) A spectral bundle method for semidefinite programming. SIAM J Optim 10(3):673–696

    Article  MathSciNet  Google Scholar 

  • Hen I, Job J, Albash T, Rønnow TF, Troyer M, Lidar DA (2015) Probing for quantum speedup in spin-glass problems with planted solutions. Phys Rev A 92(4):042325

    Article  Google Scholar 

  • Hopfield JJ, Tank DW (1985) “Neural” computation of decisions in optimization problems. Biol Cybern 52(3):141–152

    MATH  Google Scholar 

  • Inagaki T, Haribara Y, Igarashi K, Sonobe T, Tamate S, Honjo T, Marandi A, McMahon PL, Umeki T, Enbutsu K et al (2016) A coherent Ising machine for 2000-node optimization problems. Science 354(6312):603–606

    Article  Google Scholar 

  • Ising E (1925) Beitrag zur Theorie des Ferromagnetismus. Zeitschrift für Physik 31(1):253–258

    Article  Google Scholar 

  • Jensen Tommy R, Toft Bjarne (1995) Graph coloring problems. Wiley, New Jersey

    MATH  Google Scholar 

  • Johnson MW, Amin MHS, Gildert S, Lanting T, Hamze F, Dickson N, Harris R, Berkley AJ, Johansson J, Bunyk P et al (2011) Quantum annealing with manufactured spins. Nature 473(7346):194

    Article  Google Scholar 

  • Karp RM (1972) Reducibility among combinatorial problems. Complexity of computer computations. Springer, Berlin, pp 85–103

    Book  Google Scholar 

  • King AD, Lanting T, Harris R (2015) Performance of a quantum annealer on range-limited constraint satisfaction problems. arXiv:1502.02098

  • Kuramoto Y (2003) Chemical oscillations, waves and turbulence. Dover, New York

    MATH  Google Scholar 

  • Kuramoto Y (1975) Self-entrainment of a population of coupled non-linear oscillators. In: International symposium on mathematical problems in theoretical physics, pp 420–422. Springer

  • Lucas Andrew (2014) Ising formulations of many NP problems. Front Phys 2:5

    Article  Google Scholar 

  • Lucas A (2013) Ising formulations of many NP problems. arXiv:1302.5843

  • Lyapunov AM (1992) The general problem of the stability of motion. Int J Control 55(3):531–534

    Article  MathSciNet  Google Scholar 

  • Mahboob I, Okamoto H, Yamaguchi H (2016) An electromechanical Ising Hamiltonian. Sci Adv 2(6):e1600236

    Article  Google Scholar 

  • Marandi A, Wang Z, Takata K, Byer RL, Yamamoto Y (2014) Network of time-multiplexed optical parametric oscillators as a coherent Ising machine. Nat Photon 8(12):937–942

    Article  Google Scholar 

  • Martí R, Duarte A, Laguna M (2009) Advanced scatter search for the max-cut problem. INFORMS J Comput 21(1):26–38

    Article  MathSciNet  Google Scholar 

  • McMahon PL, Marandi A, Haribara Y, Hamerly R, Langrock C, Tamate S, Inagaki T, Takesue H, Utsunomiya S, Aihara K et al (2016) A fully-programmable 100-spin coherent Ising machine with all-to-all connections. Science 354:5178

    Article  Google Scholar 

  • Myklebust T (2015) Solving maximum cut problems by simulated annealing. arXiv:1505.03068

  • Neogy A, Roychowdhury J (2012) Analysis and design of sub-harmonically injection locked oscillators. In: Proceedings of IEEE DATE

  • Rinaldi G. Rudy graph generator code. https://www-user.tu-chemnitz.de/~helmberg/rudy.tar.gz. Website: https://www-user.tu-chemnitz.de/~helmberg/sdp_software.html

  • Robertson N, Sanders DP, Seymour P, Thomas R (1996) Efficiently four-coloring planar graphs. In: STOC ’96: Proceedings of the 28th annual ACM symposium on theory of computing, pp 571–575. ACM

  • Rønnow TF, Wang Z, Job J, Boixo S, Isakov SV, Wecker D, Martinis JM, Lidar DA, Troyer M (2014) Defining and detecting quantum speedup. Science 345(6195):420–424

    Article  Google Scholar 

  • Sheldon F, Traversa FL, Di Ventra M (2019) Taming a nonconvex landscape with dynamical long-range order: memcomputing Ising benchmarks. Phys Rev E 100(5):053311

    Article  Google Scholar 

  • Shinomoto S, Kuramoto Y (1986) Phase transitions in active rotator systems. Prog Theor Phys 75(5):1105–1110

    Article  Google Scholar 

  • The G-set benchmarks for MAX-CUT. http://grafo.etsii.urjc.es/optsicom/maxcut

  • Wang T, Roychowdhury J (2017) Oscillator-based Ising machine. In arXiv:1709.08102

  • Wang T, Roychowdhury J (2014) PHLOGON: PHase-based logic using oscillatory nanosystems. In: Proceedings of Unconventional Computation and Natural Computation, LNCS sublibrary: Theoretical Computer Science and General Issues. Springer

  • Wang T (2017) Sub-harmonic injection locking in metronomes. arXiv:1709.03886

  • Wang T (2017) Achieving phase-based logic bit storage in mechanical metronomes. arXiv:1710.01056

  • Wang T, Roychowdhury J (2019) OIM: Oscillator-based Ising Machines for solving Combinatorial Optimisation problems. arXiv:1903.07163

  • Wang T, Roychowdhury J (2015) Design tools for oscillator-based computing systems. In: Proceedings of IEEE DAC, pp 188:1–188:6

  • Wang T, Wu L, Roychowdhury J (2019) New Computational Results and Hardware Prototypes for Oscillator-based Ising Machines. In: Proceedings IEEE DAC, pp 239:1–239:2

  • Wang T, Roychowdhury J (2019) OIM: Oscillator-based Ising Machines for Solving Combinatorial Optimisation Problems. In: Proceedings of International Conference on Unconventional Computation and Natural Computation

  • Winfree A (1967) Biological rhythms and the behavior of populations of coupled oscillators. Theor Biol 16:15–42

    Article  Google Scholar 

  • Yamamoto K, Huang W, Takamaeda-Yamazaki S, Ikebe M, Asai T, Motomura M (2017) A time-division multiplexing Ising machine on FPGAs. In: Proceedings of the 8th International Symposium on Highly Efficient Accelerators and Reconfigurable Technologies, p 3. ACM

  • Yamaoka M, Yoshimura C, Hayashi M, Okuyama T, Aoki H, Mizuno H (2016) A 20k-spin Ising chip to solve combinatorial optimization problems with CMOS annealing. IEEE J Solid State Circ 51(1):303–309

    Article  Google Scholar 

  • Yin X, Sedighi B, Varga M, Ercsey-Ravasz M, Toroczkai Z, Hu XS (2018) Efficient analog circuits for Boolean satisfiability. IEEE Trans Very Large Scale Integr (VLSI) Syst 26(1):155–167

    Article  Google Scholar 

Download references

Acknowledgements

The authors are indebted to the anonymous reviewers for their exceptionally well-informed and thorough examination of the manuscript and their detailed, insightful suggestions for improvement. Support from the US National Science Foundation is gratefully acknowledged.

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  1. Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, USA

    Tianshi Wang, Leon Wu, Parth Nobel & Jaijeet Roychowdhury

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  1. Tianshi Wang
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Wang, T., Wu, L., Nobel, P. et al. Solving combinatorial optimisation problems using oscillator based Ising machines. Nat Comput 20, 287–306 (2021). https://doi.org/10.1007/s11047-021-09845-3

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